Chromatic and clique numbers of a class of perfect graphsMohammad RezaFanderAzad University, Chaluse Branchauthortextarticle2015eng‎Let $p$ be a prime number and $n$ be a positive integer‎. ‎The graph‎ ‎$G_p(n)$ is a graph with vertex set $[n]=\{1‎, ‎2,\ldots‎, ‎n\}$‎, ‎in‎ ‎which there is an arc from $u$ to $v$ if and only if $u\neq v$ and‎ ‎$p\nmid u+v$‎. ‎In this paper it is shown that $G_p(n)$ is a perfect‎ ‎graph‎. ‎In addition‎, ‎an explicit formula for the chromatic number of‎ ‎such graph is given‎.Transactions on CombinatoricsUniversity of Isfahan2251-86574

2015514http://toc.ui.ac.ir/article_7389_3e9ec295be34a42ee71a0570cc2fbfa9.pdfdx.doi.org/10.22108/toc.2015.7389A dynamic domination problem in treesWilliamKlostermeyerSchool of Computing
University of North FloridaauthorChristinaMynhardtDepartment of Mathematics and Statistics
University of Victoriaauthortextarticle2015eng‎We consider a dynamic domination problem for graphs in which an infinite‎ ‎sequence of attacks occur at vertices with guards and the guard at the‎ ‎attacked vertex is required to vacate the vertex by moving to a neighboring‎ ‎vertex with no guard‎. ‎Other guards are allowed to move at the same time‎, ‎and‎ ‎before and after each attack and the resulting guard movements‎, ‎the vertices‎ ‎containing guards form a dominating set of the graph‎. ‎The minimum number of‎ ‎guards that can successfully defend the graph against such an arbitrary‎ ‎sequence of attacks is the m-eviction number‎. ‎This parameter lies between the‎ ‎domination and independence numbers of the graph‎. ‎We characterize the classes of trees for which the m-eviction number equals‎ ‎the domination number and the independence number‎, ‎respectively‎.
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20151531http://toc.ui.ac.ir/article_7590_fbf0dbf66e3b8321a9266cd46dabc47a.pdfdx.doi.org/10.22108/toc.2015.7590The resistance distance and the Kirchhoff index of the $k$-th semi total point graphsDenglanCuiDepartment of Mathematics
Hunan Nornal University
Changsha, Hunan 410081authorYaopingHouDepartment of Mathematics
Hunan Normal University
Changsha, Hunan,410081authortextarticle2015eng‎The $k$-th semi-total point graph $R^k(G)$ of a graph $G$‎, ‎is a graph obtained from $G$ by adding $k$ vertices corresponding to each edge and connecting them to the endpoints of the edge considered‎. ‎In this paper‎, ‎we obtain formulas for the resistance distance and Kirchhoff index of $R^k(G).$‎
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20153341http://toc.ui.ac.ir/article_7767_d003df08d9fc1d3ea3e0f98ad110cc1b.pdfdx.doi.org/10.22108/toc.2015.7767Broadcast domination in ToriKian WeeSohDept of Mathematics, National University of SingaporeauthorKhee-MengKohDepartment of Mathematics
National University of Singaporeauthortextarticle2015engA broadcast on a graph $G$ is a function $f‎ : ‎V(G) \rightarrow \{0‎, ‎1,\dots‎, ‎diam(G)\}$ such that for every vertex $v \in V(G)$‎, ‎$f(v) \leq e(v)$‎, ‎where $diam(G)$ is the diameter of $G$‎, ‎and $e(v)$ is the eccentricity of $v$‎. ‎In addition‎, ‎if every vertex hears the broadcast‎, ‎then the broadcast is a dominating broadcast. ‎The cost of a broadcast $f$ is the value $\sigma(f) = \sum_{v \in V(G)} f(v)$‎. ‎In this paper we determine the minimum cost of a dominating broadcast (also known as the broadcast domination number) for a torus $C_{m} \;\Box\; C_{n}$‎.Transactions on CombinatoricsUniversity of Isfahan2251-86574

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20154353http://toc.ui.ac.ir/article_7654_d69d40ef7aab331d7b142121c88012e9.pdfdx.doi.org/10.22108/toc.2015.7654A classification of finite groups with integral bi-Cayley graphsMajidArezoomandDepartmant of Mathematical Sciences, Isfahan University of Technology, Isfahan, IranauthorBijanTaeriDepartment of Mathematics, Isfahan University of Technology, Isfahan, Iranauthortextarticle2015engThe bi-Cayley graph of a finite group $G$ with respect to a subset $S\subseteq G$‎, ‎which is denoted by $BCay(G,S)$‎, ‎is the graph with‎ ‎vertex set $G\times\{1,2\}$ and edge set $\{\{(x,1)‎, ‎(sx,2)\}\mid x\in G‎, ‎\ s\in S\}$‎. ‎A‎ ‎finite group $G$ is called a \textit{bi-Cayley integral group} if for any subset $S$ of‎ ‎$G$‎, ‎$BCay(G,S)$ is a graph with integer eigenvalues‎. ‎In this paper we prove‎ ‎that a finite group $G$ is a bi-Cayley integral group if and only if $G$ is isomorphic to‎ ‎one of the groups $\Bbb Z_2^k$‎, ‎for some $k$‎, ‎$\Bbb Z_3$ or $S_3$‎.Transactions on CombinatoricsUniversity of Isfahan2251-86574